A lattice point is a point with integer coordinates such as (2,3).In how many ways can we pick 3 lattice points such that both coordinates of all three points are nonnegative integers less than 4, and connecting the three points forms a triangle with positive area?

I'll give this one a shot......but....I'm not overly confident.....!!!

Notice the follwing patterns......

For a 1 x 1 square, we have two possible values fo x and y, 0 and 1. And we have two ways to pick an apex point and a base of 1. But, we can orient these two triangles in 4 ways......apex points at the "top," apex points to the "right," apeax points to the "left" and apex points on the "bottom"....and each will have a base of 1...so that's eight possible triangles

For a 2 x 2 square, we have three possible values for x and y .... 0, 1, 2. And we have three possible apex points, three possible bases - two of one unit and one of two units, and three possible heights - two of one unit and one of two units. And these can be oriented in 4 different ways, as before.

We have four possible values for x and y - 0,1,2,3. And we have four possible apex points, 6 possible bases and heights- three of one unit, two of two units and three of one unit.... and four possible orientations, as before.

Yep.....after reconsidering....I don't like my answer at all..I realize there might be some double - if not "triple" counting in mine....I'll look at Melody's answer in more detail......but.....just not right now.....

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